Question:
If $\int \frac{dx}{x+x^{7}} = p\left(x\right)$ then, $\int \frac{x^{6}}{x+x^{7}}dx$ is equal to:
If $\int \frac{dx}{x+x^{7}} = p\left(x\right)$ then, $\int \frac{x^{6}}{x+x^{7}}dx$ is equal to:
Updated On: Nov 24, 2025
- $In \left|x\right| -p\left(x\right) + c$
- $In \left|x\right| +p\left(x\right) + c$
- $x-p \left(x\right) + c$
- $x+p \left(x\right) + c$
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The Correct Option is A
Solution and Explanation
$\int \frac{x^{6}}{x+x^{7}}dx = \int \frac{x^{6}}{x\left(1+x^{6}\right)}dx$
$= \int \frac{\left(1+x^{6}\right)-1}{x\left(1+x^{6}\right)}dx$
$=\int \frac{1}{x}dx-\int \frac{1}{x+x^{7}}dx$
$= In \left|x\right| -p\left(x\right) + c$
$= \int \frac{\left(1+x^{6}\right)-1}{x\left(1+x^{6}\right)}dx$
$=\int \frac{1}{x}dx-\int \frac{1}{x+x^{7}}dx$
$= In \left|x\right| -p\left(x\right) + c$
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
